. In testing a new drug, researchers found that 20% of all patients using it will have a mild side effect. A random sample of 15 patients using the drug is selected. Find the probability that:
(A) exactly three will have this mild side effect
(B) at least six will have this mild side effect
See your other post on this problem.
To find the probability of these events, we need to use the binomial probability formula. The formula for a binomial distribution is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes in n trials
- (n choose k) represents the number of ways to choose k successes out of n trials
- p is the probability of success on a single trial
- (1-p) is the probability of failure on a single trial
- k is the number of successes we want to find the probability for
- n is the total number of trials or observations
(A) To find the probability that exactly three patients will have the mild side effect, we have:
- n = 15 (the total number of patients in the sample)
- k = 3 (the number of patients with the mild side effect)
- p = 0.20 (the probability of having the mild side effect for a single patient)
Using the binomial probability formula:
P(X = 3) = (15 choose 3) * 0.20^3 * (1 - 0.20)^(15 - 3)
We can calculate this probability using a calculator or software that can compute binomial probabilities.
(B) To find the probability that at least six patients will have the mild side effect, we need to calculate the probabilities of having six, seven, eight, ..., up to fifteen patients with the mild side effect, and then sum them up.
P(X >= 6) = P(X = 6) + P(X = 7) + ... + P(X = 15)
You can use the binomial probability formula as mentioned above to calculate each individual probability and then sum them up.
Alternatively, you can use the complement rule to find the probability of at least six patients having the mild side effect. The complement of "at least six" is "less than six," so you can calculate:
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
And then subtract this probability from 1 to get the probability of at least six patients having the mild side effect:
P(X >= 6) = 1 - P(X < 6)